Asymptotic results on weakly increasing subsequences in random words

Abstract

Let X = (X1, . . . , Xn) be a vector of i.i.d. random variables where Xi's take values over N. The purpose of this paper is to study the number of weakly increasing subsequences of X of a given length k, and the number of all weakly increasing subsequences of X. For the former, it is shown that a central limit theorem holds. Also, the first two moments of each of those two random variables are analyzed, their asymptotics are investigated, and results are related to the case of similar statistics in uniformly random permutations. We conclude the paper with applications on a similarity measure of Steele, and on increasing subsequences of riffle shuffles.

A study on random permutation graphs

“…Let us note that the literature on the number of increasing subsequences of a random permutation is vast. The central limit just given was proven for random permutations as well as random words was proven in [24]. We will mention a few more pointers here for an analysis of the number of cycles of a random permutation graph.…”

“…In other words, K m is merely the number of decreasing subsequences of length m in π n . Noting that this equals in distribution to the number of increasing subsequences of length m in π n , and denoting the latter by I n,m , [24] shows that…”

“…Now we move on to proving the central limit theorem. Although a brief sketch for the number of increasing subsequences of a given length in a random permutation was given in [24], we include all the details here for the sake of completeness 1 . First, the proof is based on U -statistics, and a general reference for such statistics is [25].…”

“…The reason why it was only a sketch in[24] is that a similar argument on random words was detailed there.…”

On unfair permutations

“…The discussion on number of inversions can be generalized to increasing (or decreasing) sequences of arbitrary length. This statistic in uniformly random permutation framework was previously studied in [5]. Their proof is a lot simpler due to underlying symmetry.…”